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I need to compute the coefficient for the Holder continuity of $x^p$ with $x > 0$, that is $$ H(p) := \sup_{x\neq y}\frac{|x^p - y^p|}{|x - y|^p}. $$ I am actually going to apply this in numerical scheme, so I am interested in finding $H(p)$ itself, rather than upper bounds, or at least an upper bound which is not extremely conservative. I am also interested in case of $H(p,M)$ which is the Holder coefficient over a bounded interval $[0, M]$. Any hints how to do that?

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Taking $y=0$ shows that $H(p)\ge1$. If $0<p\le1$, the inequality $1-z^p\le(1-z)^p$ for $0\le z\le1$ shows that $H(p)=1$. If $p>1$, choosing $x=y+1$ shows that $H(p)=\infty$. The same happens on a bounded interval $[0,M]$.

As an aside, observe that if $$ |f(x)-f(y)|\le C\,|x-y|^r $$ with $r>1$, then $f$ is constant.

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